The word certainly predates the motor car, and horse-drawn carts and carriages had dashboards which were:
A board or leathern apron in the front of a vehicle, to prevent mud from being splashed by the heels of the horses upon the interior of the vehicle. Also, movable sides to a cart for the same purpose (Halliwell). (OED)
Doubtless this is where the name comes from. But why were they called dashboards? The verb dash means to strike or hit. Was it that the earliest dashboard was just a hit board for mud, stones etc?